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In this paper we show that a complete and non-compact surface immersed in theEuclidean space with quadratic extrinsic area growth has finite total curvatureprovided the surface has tamed second fundamental form and admits totalcurvature. In such a case we obtain as well a generalized Chern-Ossermaninequality. In the particular case of a surface of nonnegative curvature, weprove that the surface is diffeomorphic to the Euclidean plane if the surfacehas tamed second fundamental form, and that the surface is isometric to theEuclidean plane if the surface has strongly tamed second fundamental form. Inthe last part of the paper we characterize the fundamental tone of anysubmanifold of tamed second fundamental form immersed in an ambient space witha pole and quadratic decay of the radial sectional curvatures.

On the total curvature and extrinsic area growth of surfaces with tamed second fundamental form